Multiple choice

Find the quadratic equation in $x$, whose roots are $(m + n)$ and $(m - n)$

  1. $x^{2} + 2mx + (m^{2} + n^{2}) = 0$
  2. $x^{2} - 2mx + (m^{2} - n^{2}) = 0$
  3. $x^{2} - mx + (m^{2} - n^{2}) = 0$
  4. $x^{2} - mnx + (m^{2} - n^{2}) = 0$
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B Correct answer
Explanation

A quadratic equation with roots r1 and r2 is given by x^2 - (r1+r2)x + (r1*r2) = 0. Here, r1+r2 = (m+n) + (m-n) = 2m, and r1*r2 = (m+n)(m-n) = m^2 - n^2. Substituting these gives x^2 - 2mx + (m^2 - n^2) = 0.

AI explanation

Using the relationships between roots and coefficients, the sum of the roots is (m+n) + (m-n) = 2m and the product is (m+n)(m-n) = m^2 - n^2. The required quadratic equation is x^2 - (sum)x + (product) = 0. Substituting the calculated values gives x^2 - 2mx + (m^2 - n^2) = 0.

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